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Your data matches 171 different statistics following compositions of up to 3 maps.
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Matching statistic: St000392
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00109: Permutations —descent word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => 0 => 0
[2,1] => [1,2] => 0 => 0
[1,2,3] => [1,2,3] => 00 => 0
[1,3,2] => [1,3,2] => 01 => 1
[2,1,3] => [1,3,2] => 01 => 1
[2,3,1] => [1,2,3] => 00 => 0
[3,1,2] => [1,2,3] => 00 => 0
[3,2,1] => [1,2,3] => 00 => 0
[1,2,3,4] => [1,2,3,4] => 000 => 0
[1,2,4,3] => [1,2,4,3] => 001 => 1
[1,3,2,4] => [1,3,2,4] => 010 => 1
[1,3,4,2] => [1,3,4,2] => 001 => 1
[1,4,2,3] => [1,4,2,3] => 010 => 1
[1,4,3,2] => [1,4,2,3] => 010 => 1
[2,1,3,4] => [1,3,4,2] => 001 => 1
[2,1,4,3] => [1,4,2,3] => 010 => 1
[2,3,1,4] => [1,4,2,3] => 010 => 1
[2,3,4,1] => [1,2,3,4] => 000 => 0
[2,4,1,3] => [1,3,2,4] => 010 => 1
[2,4,3,1] => [1,2,4,3] => 001 => 1
[3,1,2,4] => [1,2,4,3] => 001 => 1
[3,1,4,2] => [1,4,2,3] => 010 => 1
[3,2,1,4] => [1,4,2,3] => 010 => 1
[3,2,4,1] => [1,2,4,3] => 001 => 1
[3,4,1,2] => [1,2,3,4] => 000 => 0
[3,4,2,1] => [1,2,3,4] => 000 => 0
[4,1,2,3] => [1,2,3,4] => 000 => 0
[4,1,3,2] => [1,3,2,4] => 010 => 1
[4,2,1,3] => [1,3,2,4] => 010 => 1
[4,2,3,1] => [1,2,3,4] => 000 => 0
[4,3,1,2] => [1,2,3,4] => 000 => 0
[4,3,2,1] => [1,2,3,4] => 000 => 0
[1,2,3,4,5] => [1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => [1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => [1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => [1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => [1,2,5,3,4] => 0010 => 1
[1,2,5,4,3] => [1,2,5,3,4] => 0010 => 1
[1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => [1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => [1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => [1,3,5,2,4] => 0010 => 1
[1,3,5,4,2] => [1,3,5,2,4] => 0010 => 1
[1,4,2,3,5] => [1,4,2,3,5] => 0100 => 1
[1,4,2,5,3] => [1,4,2,5,3] => 0101 => 1
[1,4,3,2,5] => [1,4,2,5,3] => 0101 => 1
[1,4,3,5,2] => [1,4,2,3,5] => 0100 => 1
[1,4,5,2,3] => [1,4,5,2,3] => 0010 => 1
[1,4,5,3,2] => [1,4,5,2,3] => 0010 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000298
(load all 43 compositions to match this statistic)
(load all 43 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
St000298: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
Description
The order dimension or Dushnik-Miller dimension of a poset.
This is the minimal number of linear orderings whose intersection is the given poset.
Matching statistic: St000846
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00013: Binary trees —to poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> 1 = 0 + 1
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> 1 = 0 + 1
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> 1 = 0 + 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> 2 = 1 + 1
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> 2 = 1 + 1
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 2 = 1 + 1
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000183
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000183: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2]
=> []
=> 0
[2,1] => [1,2] => [2]
=> []
=> 0
[1,2,3] => [1,2,3] => [3]
=> []
=> 0
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,3,1] => [1,2,3] => [3]
=> []
=> 0
[3,1,2] => [1,2,3] => [3]
=> []
=> 0
[3,2,1] => [1,2,3] => [3]
=> []
=> 0
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> 0
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> 0
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> 0
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> 0
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> 0
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> 0
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> 0
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St000225
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000253
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000253: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[2,1] => [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 0
[1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[1,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,1,3] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 0
[3,1,2] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[3,2,1] => [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
[1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,2,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,3,2,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[1,4,2,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[1,4,3,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[2,1,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[2,1,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[2,3,1,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 0
[2,4,1,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[2,4,3,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
[3,1,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,1,4,2] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[3,2,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,2,4,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 1
[3,4,1,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[3,4,2,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1
[4,1,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,1,3,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,2,1,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,2,3,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,3,1,2] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[4,3,2,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
Description
The crossing number of a set partition.
This is the maximal number of chords in the standard representation of a set partition, that mutually cross.
Matching statistic: St000319
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000442
Mp00223: Permutations —runsort⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00012: Binary trees —to Dyck path: up step, left tree, down step, right tree⟶ Dyck paths
St000442: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[2,1] => [1,2] => [.,[.,.]]
=> [1,0,1,0]
=> 0
[1,2,3] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,3,2] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,1,3] => [1,3,2] => [.,[[.,.],.]]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,1,2] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[3,2,1] => [1,2,3] => [.,[.,[.,.]]]
=> [1,0,1,0,1,0]
=> 0
[1,2,3,4] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,3,4,2] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,4,2,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,4,3,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,3,4] => [1,3,4,2] => [.,[[.,[.,.]],.]]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,1,4,3] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,3,4,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,4,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,4,3,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,2,4] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,1,4,2] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,1,4] => [1,4,2,3] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[3,2,4,1] => [1,2,4,3] => [.,[.,[[.,.],.]]]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,4,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,4,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,1,3,2] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,1,3] => [1,3,2,4] => [.,[[.,.],[.,.]]]
=> [1,0,1,1,0,0,1,0]
=> 1
[4,2,3,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,1,2] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[4,3,2,1] => [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1
[1,2,4,3,5] => [1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,2,5,3,4] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,5,4,3] => [1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,3,4,2,5] => [1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,4,5,2] => [1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,3,5,2,4] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,3,5,4,2] => [1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,2,5,3] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,2,5] => [1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1
[1,4,3,5,2] => [1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,4,5,3,2] => [1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
Description
The maximal area to the right of an up step of a Dyck path.
Matching statistic: St000480
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000480: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [1,2] => [1,1]
=> 0
[2,1] => [1,2] => [1,2] => [1,1]
=> 0
[1,2,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[1,3,2] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,1,3] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[2,3,1] => [1,2,3] => [1,2,3] => [1,1,1]
=> 0
[3,1,2] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[3,2,1] => [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,4,2,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,4,3,2] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 0
[2,4,1,3] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[2,4,3,1] => [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,1,4,2] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,2,4,1] => [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 1
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[4,1,2,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,1,3] => [1,4,3,2] => [1,4,3,2] => [2,1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,1,2] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[4,3,2,1] => [1,4,2,3] => [1,4,3,2] => [2,1,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,3,5,2,4] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [1,2,3,5,4] => [2,1,1,1]
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,2,5,3] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,3,2,5] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,3,5,2] => [1,2,4,5,3] => [1,2,5,4,3] => [2,1,1,1]
=> 1
[1,4,5,2,3] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,2,4,3,5] => [1,2,4,3,5] => [2,1,1,1]
=> 1
Description
The number of lower covers of a partition in dominance order.
According to [1], Corollary 2.4, the maximum number of elements one element (apparently for $n\neq 2$) can cover is
$$
\frac{1}{2}(\sqrt{1+8n}-3)
$$
and an element which covers this number of elements is given by $(c+i,c,c-1,\dots,3,2,1)$, where $1\leq i\leq c+2$.
The following 161 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000535The rank-width of a graph. St000552The number of cut vertices of a graph. St000659The number of rises of length at least 2 of a Dyck path. St000845The maximal number of elements covered by an element in a poset. St000897The number of different multiplicities of parts of an integer partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001280The number of parts of an integer partition that are at least two. St001333The cardinality of a minimal edge-isolating set of a graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001354The number of series nodes in the modular decomposition of a graph. St001393The induced matching number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001512The minimum rank of a graph. St001587Half of the largest even part of an integer partition. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001743The discrepancy of a graph. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000013The height of a Dyck path. St000147The largest part of an integer partition. St000159The number of distinct parts of the integer partition. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000259The diameter of a connected graph. St000306The bounce count of a Dyck path. St000381The largest part of an integer composition. St000396The register function (or Horton-Strahler number) of a binary tree. St000397The Strahler number of a rooted tree. St000444The length of the maximal rise of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000668The least common multiple of the parts of the partition. St000679The pruning number of an ordered tree. St000701The protection number of a binary tree. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000758The length of the longest staircase fitting into an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000783The side length of the largest staircase partition fitting into a partition. St000903The number of different parts of an integer composition. St001062The maximal size of a block of a set partition. St001093The detour number of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001128The exponens consonantiae of a partition. St001261The Castelnuovo-Mumford regularity of a graph. St001432The order dimension of the partition. St001674The number of vertices of the largest induced star graph in the graph. St001732The number of peaks visible from the left. St000318The number of addable cells of the Ferrers diagram of an integer partition. St000759The smallest missing part in an integer partition. St000299The number of nonisomorphic vertex-induced subtrees. St000640The rank of the largest boolean interval in a poset. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000618The number of self-evacuating tableaux of given shape. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000058The order of a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St000451The length of the longest pattern of the form k 1 2. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000260The radius of a connected graph. St001734The lettericity of a graph. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000094The depth of an ordered tree. St000862The number of parts of the shifted shape of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000485The length of the longest cycle of a permutation. St000456The monochromatic index of a connected graph. St001665The number of pure excedances of a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001737The number of descents of type 2 in a permutation. St001877Number of indecomposable injective modules with projective dimension 2. St000454The largest eigenvalue of a graph if it is integral. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St001729The number of visible descents of a permutation. St000542The number of left-to-right-minima of a permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000308The height of the tree associated to a permutation. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St000387The matching number of a graph. St000486The number of cycles of length at least 3 of a permutation. St001071The beta invariant of the graph. St001271The competition number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001638The book thickness of a graph. St001728The number of invisible descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000166The depth minus 1 of an ordered tree. St000258The burning number of a graph. St000328The maximum number of child nodes in a tree. St000452The number of distinct eigenvalues of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000918The 2-limited packing number of a graph. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001530The depth of a Dyck path. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St001112The 3-weak dynamic number of a graph. St001110The 3-dynamic chromatic number of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001330The hat guessing number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000662The staircase size of the code of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000307The number of rowmotion orbits of a poset. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001555The order of a signed permutation. St001488The number of corners of a skew partition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000632The jump number of the poset. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001645The pebbling number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001896The number of right descents of a signed permutations. St001624The breadth of a lattice. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001820The size of the image of the pop stack sorting operator. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001823The Stasinski-Voll length of a signed permutation. St001946The number of descents in a parking function. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001935The number of ascents in a parking function. St001960The number of descents of a permutation minus one if its first entry is not one. St001569The maximal modular displacement of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001574The minimal number of edges to add or remove to make a graph regular. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001742The difference of the maximal and the minimal degree in a graph. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001570The minimal number of edges to add to make a graph Hamiltonian.
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